Point transformations and renormalization in the unitary gauge

It is shown by means of a model that the renormalization and unitary gauges can be connected by a point transformation, and this fact is used to construct a formal proof of renormalization in the unitary gauge. The formal proof is then verified by demonstrating that for a fourth-order on-shell scattering process the S-matrix calculated directly in the unitary gauge is exactly equal to that calculated in the renormalization gauge. The calculation is refined to the point where it becomes purely graphical and this allows one to see by inspection how the cancellation of divergences occurs in the unitary gauge. The model considered here is Abelian, but it will be generalized to the non-Abelian case subsequently.